MATLAB: How to get a Fourier-limited Gaussian pulse

fftgausspulse

Hi all, I have generated a femtosecond Gauss pulse and fft it to get the frequency spectrum. But when I calculate the time bandwidth product, it is never a Fourier limited pulse, e.g., the time duration (FWHM) is 20fs, and the bandwidth is around 0.2eV(read from the graph), and 20fs*0.2eV is never close to the theoretical limit 1.825 eV*fs. In principle, this Gaussian pulse be Fourier limited. I don't know if I make some mistakes here. I hope that someone can help me with this. Thanks! Below is my code:
t=[-100:0.01:100];
A=0.1519;
T0=20;   %FWHM of the pulse
w=2*pi*300/800;   %frequency centered at 800nm
y=A.*exp(-4*log(2)*(t/T0).^2).*cos(w.*t);  %Gauss pulse in terms of FWHM expression
figure;plot(t,y);
Y=fftshift(fft(y));
Y_mag=abs(Y);
X=(-(length(Y_mag)-1)/2:(length(Y_mag)-1)/2)*1239.84/300/length(Y_mag)/(t(2)-t(1));  %frequency axis
figure;plot(X,Y_mag);

Best Answer

Hi Andi,
The issue here is that for a voltage pulse the fourier transform is done on the linear voltage function y(t) to produce Y(f), but the FWHM condition is on the square of y(t) and Y(f). I modified your code to produce a gaussian pulse y(t) as the square root of your y(t) [which in this context is actually y(t)^2]. Then keep the cos(wt) carrier, do the fft to get Y(E), and plot y^2 and Y^2 for FWHM. The height of the peaks don't matter for FWHM, so I made no attempt to figure out the constants in front.
For the Y(E)^2 plot, the FWHM of each peak is about .09 eV (it's a bit undersampled), and 20fs times that is 1.8, so it works.
I added figure numbers since I'm not a fan of new figures popping up every time you rerun the code.
t=-100:0.01:100;
A=0.1519;
T0=20;   %FWHM of the pulse
w=2*pi*300/800;   %frequency centered at 800nm
y=sqrt(exp(-4*log(2)*(t/T0).^2)).*cos(w.*t);  %Gauss pulse in terms of FWHM expression
y2 = y.^2;
figure(1)
plot(t,y2);
Y=fftshift(fft(y));
Y2_mag=abs(Y.^2);
X=(-(length(Y_mag)-1)/2:(length(Y_mag)-1)/2)*1239.84/300/length(Y_mag)/(t(2)-t(1));  %frequency axis
figure(2)
plot(X,Y2_mag,'-o');
xlim([1.4 1.7])
ylim([0 3e6])
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